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Critical Numbers and Points of Inflection
Rolle's Theorem/Mean Value Theorem
First and Second Derivative Tests
Curve Sketching
Mystery
100

In order to find critical numbers, you need to find where the derivative is equivalent to these.

What is zero and undefined?

100

Let f be continuous on the closed interval [a, b] and differentiable on the open interval (a, b). If f(a) = f(b) then this is guaranteed to exist for at least one number c in (a, b) such that this is true.

What is f'(c) = 0?

100

This is what the First Derivative Test is useful in locating.

What are local (relative) extrema, increasing intervals, and decreasing intervals?

100
These are the characteristics of a function on an interval when f ' (x) is positive and f '' (x) is negative.
What is increasing and concave down?
100
Along with identifying all given quantities and quantities to be determined, this is part of the first step in solving optimization problems.
What is draw a picture?
200

This is what occurs on the second derivative if the original function has a point of inflection.

What is changes between positive and negative?

200

The number of zeros (roots) of f(x)=x3+4x+1 on the interval [-1, 1].

What is one?

200

This is what the Second Derivative Test is useful in locating.

What are inflection points and intervals of concavity?

200
This characteristic of f(x) is true when f ' (x) is increasing.
What is concave up?
200
The function you are attempting to optimize is called this.
What is the objective function?
300

This is what occurs on the first derivative if the original function has a point of inflection.

What is changes between increasing and decreasing?

300

If f is continuous on the closed interval [a, b] and differentiable on the open interval (a, b), then there exists a number c in (a, b) such that this is true.

What is f'(c) = [f(b) - f(a)]/[b - a].

300

This is the name for the values that are possible extrema.

What are critical numbers.

300
These are the equations of the horizontal asymptotes for the function f(x) = sqrt(4x^2+1)/(x+1).
What are y=2 and y= -2 ?
300
This is the main reason for finding a constraint equation.
What is to rewrite the objective function in terms of a single variable?
400

These are the critical numbers of f(x) = x2(x2 - 4).

What are x=0, 2, -2?

400

Let f(x) = x4 - 2x2. These are the values of c in the interval (-2, 2) guarenteed by Rolle's Theorem.

What are 0, 1, and -1?

400

These are the local extrema of f(x) = (x2 - 4)2/3.


What is a relative minimum at (-2, 0) and (2, 0) and a relative maximum at (0, 161/3)?

400
The interval(s) on which y = x^3 - 12x is increasing.
What is (-infinity, -2) U (2, infinity)?
400
This is the point on f(x) = x^2 that is closest to (2, 1/2).
What is (1, 1)?
500

These are the points of inflection of f(x) = x4 - 4x3

What are (0,0) and (2,-16)?

500

Let f(x) = x2/3 on [0, 1]. This is the value of c that is guaranteed by the Mean Value Theorem.

What is 8/27?

500

These are the relative extrema for f(x) = -3x5 + 5x3.

What is a local minimum at (-1, -2) and a local maximum at (1, 2)?

500

This is the interval on which f(x)=2x2/(x2-1) is concave down, given that f''(x) = (12x2+4)/(x2-1)^3.

What is (-1,1)?

500
A farmer plans to fence a rectangular pasture adjacent to a river. The pasture must contain 180,000 square meters in order to provide enough grass for the herd and does not need fencing along the river. These are the dimensions required to use the least amount of fencing.
What is 600 X 300 meters?